Class Notes for Math 921/922: Real Analysis, Instructor Mikil Foss

Topics include: Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon- Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration, Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon-Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration.

Summarization in Math Class

This action research study of twenty students in my sixth grade mathematics classroom examines the implementation of summarization strategies. Students were taught how to summarize concepts and how to explain their thinking in different ways to the teacher and their peers. Through analysis of students’ summaries of concepts from lessons that I taught, tests scores, and student journals and interviews, I discovered that summarizing mathematical concepts offers students an engaging opportunity to better understand those concepts and render that understanding more visible to the teacher. This analysis suggests that non-traditional summarization, such as verbal and written strategies, and strategies involving movement and discussions, can be useful in mathematics classrooms to improve student understanding, engagement in learning tasks, and as a form of formative assessment.

Review of Applied Spatial Data Analysis with R by R. S. Bivand, E. J. Pebesma, and V. Gomez-Rubio

Most authors struggle to pick a title that adequately conveys all of the material covered in a book. When I first saw Applied Spatial Data Analysis with R, I expected a review of spatial statistical models and their applications in packages (libraries) from the CRAN site of R. The authors’ title is not misleading, but I was very pleasantly surprised by how deep the word “applied” is here. The first half of the book essentially covers how R handles spatial data. To some statisticians this may be boring. Do you want, or need, to know the difference between S3 and S4 classes, how spatial objects in R are organized, and how various methods work on the spatial objects? A few years ago I would have said “no,” especially to the “want” part. Just let me slap my EXCEL spreadsheet into R and run some spatial functions on it. Unfortunately, the world is not so simple, and ultimately we want to minimize effort to get all of our spatial analyses accomplished. The first half of this book certainly convinced me that some extra effort in organizing my data into certain spatial class structures makes the analysis easier and less subject to mistakes. I also admit that I found it very interesting and I learned a lot.

The Economics of Threatened Species Conservation: A Review and Analysis

Stabilizing human population size and reducing human-caused impacts on the environment are keys to conserving threatened species (TS). Earth's human population is ~ 7 billion and increasing by ~ 76 million per year. This equates to a human birth-death ratio of 2.35 annually. The 2007 Red List prepared by the International Union for Conservation of Nature and Natural Resources (IUCN) categorized 16,306 species of vertebrates, invertebrates, plants, and other organisms (e.g., lichens, algae) as TS. This is ~ 1 percent of the 1,589,161 species described by IUCN or ~ 0.0033 percent of the believed 5,000,000 total species. Of the IUCN’s described species, vertebrates comprised relatively the most TS listings within respective taxonomic categories (5,742 of 59,811), while invertebrates (2,108 of 1,203,175), plants (8,447 of 297,326), and other species (9 of 28,849) accounted for minor class percentages. Conservation economics comprises microeconomic and macroeconomic principles involving interactions among ecological, environmental, and natural resource economics. A sustainable-growth (steady-state) economy has been posited as instrumental to preserving biological diversity and slowing extinctions in the wild, but few nations endorse this approach. Expanding growth principles characterize most nations' economic policies. To date, statutory fine, captive breeding cost, contingent valuation analysis, hedonic pricing, and travel cost methods are used to value TS in economic research and models. Improved valuation methods of TS are needed for benefit-cost analysis (BCA) of conservation plans. This Chapter provides a review and analysis of: (1) the IUCN status of species, (2) economic principles inherent to sustainable versus growth economies, and (3) methodological issues which hinder effective BCAs of TS conservation.